This paper has some interesting observations and the Lanczos trick is very nice, but the conclusion is no good.
I will paraphrase the argument in the paper as follows:
The naturalness and hierarchy problems are caused by perturbation theory diseases. If you have a hierarchy put in by hand, it doesn't matter if you add a high-energy sector, you can keep the hierarchy stable even if perturbation theory says it is unstable.
The reason the author says this is because he estimates how perturbation theory shifts eigenvalues in an exactly diagonalizable matrix, where he knows the answer, and he shows that the shift is not reliably given by perturbation theory. This is absolutely correct, and it shows that the perturbative argument for the shift in the Higgs mass away from zero is not persuasive by itself. If this were the real argument for hierarchy, this paper would show you that there is no argument.
But this is not the real argument. The real argument is Wilsonian.
The Real Hierarchy Problem
consider any regulator for the standard model, like a lattice. The chiral fields (the fermions and quarks) are naturally massless, and have an infinite Compton wavelength compared to the lattice, so they are always automatically tuned to their critical point. This is actually not so simple to understand, because it requires some crazy manipulations to get chiral fermions to naturally emerge with a lattice cutoff. But in this case, the lattice cutoff is just a stand-in for some sort of string-scale regularization, where chiral fermions emergy naturally.
The gauge fields are naturally massless, so they are also self-organized to their critical point, except that the SU(3) is confining, which picks out a confinement scale which is as the log of the lattice-scale coupling, so it is enormous even when the coupling is not terribly small. Anyway, everything is critical, and we would see a gravity-free version of something like our universe at long distances in this model.
Except there is no Higgs. Now if you introduce a lattice Higgs sector, if it is a fundamental scalar and not a composite of fermions, you get into trouble. The Higgs is not massless, and your lattice standard model will have a Higgs field correlation function which falls off order one with unit of length equal to the lattice spacing. It won't help to set the lattice mass term of the Higgs to zero. The quartic term means that the lattice mass has to be some negative specific value to reach the massless Higgs point since the phase transition point is not at 0 when you have a quartic self-coupling.
The reason is that the Higgs itself is like the Ising model--- it has a tunable $m^2$ (which can be negative or positive, despite the name), and you need to make $m^2$ negative order 1 unit on the lattice to reach the critical point in the Ising model. In the standard model, the critical point will be somewhere else, since it depends on the details of the fields you have in the theory, but it will not be at zero, it will be at some crazy value that depends both on the field content and on the regulator.
Yet we see that the Higgs is critical in terms of the regulator. This is the ridiculous fine-tuning which is called the Hierarchy problem--- why is the Higgs $m^2$ term fine-tuned on the lattice scale to give a nearly massless Higgs? Why is the ising-like Higgs sector look like an Ising model at the critical point, rather than an Ising model at a generic point, where the SU(2) and U(1) fields would acquire lattice-scale masses from the Higgs VEV (or else remain unbroken, if the Higgs is stable at zero field)?
This order 1 shift at the lattice scale is what is nonperturbatively tantamount to a quadratically divergent perturbative divergent shift in the Higgs mass. An order 1 shift at the lattice scale is an order $\Lambda^2$ shift when you are looking at the thing in units appropriate for the low-energy theory. This shift is absolutely real--- you can see it by lattice simulating, and it is not fixed in the standard model. There is always an order 1 shift.
In SUSY models, the Higgs mass is partnered with a chiral fermion mass. So keeping one SUSY unbroken means that you only get as much Higgs mass as there is SUSY breaking. This has nothing to do with perturbation theory. It's the location of the phase transition point in the Wilsonian point of view.
If you say "The Higgs is just fine-tuned to the phase transition point on the lattice", you are being absurd. Such a fine-tuning to a critical point screams for a dynamical mechanism. You might say "It's anthropic", but even after you tune the Higgs, as far as we can see, you need anthropic to tune the cosmological constant, and it is a little ridiculous to claim two parameters are turned anthropically to be so much smaller than Planckian, especially when there are coincidences that need to be explained:
- The Higgs scale is only 3 orders of magnitude bigger than the QCD scale, but 14 orders of magnitude less than the Planck/GUT scale. Why is there a QCD Higgs near coincidence? This would be explained by technicolor, if the couplings start off the same for some other gauge group, and the confinement just happens a little sooner because of quicker running.
- The cosmological constant scale is as far lower than the Higgs scale as the Higgs scale is lower than the Planck scale. This could be a coincidence, but it feels like a clue.
It is not reasonable to claim that this kind of structure is a random anthropic coincidence. There is more information there than in a random anthropic tuning.
However, the paper is right that the arguments for SUSY are extremely weak. Stabilizing the Higgs can be done in a variety of ways, and SUSY is only preferred because it is the one that is suggested by string theory. The fact that strings were out when SUSY was formulated meant that people made up other reasons to like SUSY, but really, the only thing that suggests SUSY is strings.
But even though we know that the fundamental theory is SUSY, that doesn't mean that the low-energy theory has to be SUSY, but it does allow the low-energy theory to be SUSY, and this does stabilize the hierarchy problem, which is a real problem.
